| L(s) = 1 | − 9.20i·3-s − 204.·5-s − 264.·7-s + 644.·9-s + 521.·11-s − 2.98e3i·13-s + 1.87e3i·15-s + 3.44e3·17-s + (6.83e3 + 513. i)19-s + 2.43e3i·21-s + 1.10e3·23-s + 2.60e4·25-s − 1.26e4i·27-s + 3.96e4i·29-s − 5.38e4i·31-s + ⋯ |
| L(s) = 1 | − 0.340i·3-s − 1.63·5-s − 0.770·7-s + 0.883·9-s + 0.391·11-s − 1.36i·13-s + 0.556i·15-s + 0.700·17-s + (0.997 + 0.0748i)19-s + 0.262i·21-s + 0.0904·23-s + 1.66·25-s − 0.642i·27-s + 1.62i·29-s − 1.80i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0748i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.997 - 0.0748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4663681395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4663681395\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-6.83e3 - 513. i)T \) |
| good | 3 | \( 1 + 9.20iT - 729T^{2} \) |
| 5 | \( 1 + 204.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 264.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 521.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 2.98e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.44e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 1.10e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 3.96e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 5.38e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 4.69e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 4.12e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 8.24e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 5.59e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.23e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.76e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.26e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.24e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 6.01e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.12e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.22e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.56e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 6.96e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 9.82e5iT - 8.32e11T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.25682448349519821560108927527, −9.273386911297115282477254995407, −7.88730798327967095914116465010, −7.55944377718560551878803250964, −6.48571938939185523439120661495, −5.08557378139462452043074989350, −3.78404508762572315579935584128, −3.13904526015408324512729465551, −1.12343529241771105030611427059, −0.14100712606660501201106180088,
1.27203653655265452983036573216, 3.24424609314664788766310154862, 3.97335186891862413445365262058, 4.85927267813050407593220211699, 6.59804571602008653265887924495, 7.25581740188069384228714522153, 8.271927114725114105268826300650, 9.395706133100313895889794969839, 10.11105344287930360690949933354, 11.41823863335905750106390119395
